Ultrafast Electron Diffraction. 4. Molecular Structures and Coherent

2766

J. Phys. Chem. 1994,98, 2766-2781

ARTICLES Ultrafast Electron Diffraction. 4. Molecular Structures and Coherent Dynamics J. Charles Williamsont and Ahmed H. Zewail' Arthur Amos Noyes Laboratory of Chemical Physics,$ Calijornia Institute of Technology, Pasadena, California 91 125 Received: January 13, 1994'

Ultrafast electron diffraction (UED) is developed, in this and the accompanying paper, as a method for studying gas-phase molecular structure and dynamics on the picosecond (ps) to femtosecond (fs) time scale. Building on our earlier reports (henceforth referred to as 1-3), we discuss theoretical and experimental considerations for the approach. Specifically we show that the use of rotational and vibrational coherences can add a new dimension to structural determination of gas-phase species. In addition to the internuclear separations of the molecular sample, the spatial alignment reflected in the scattering pattern contains bond angles and rotational constants for both excited-state and ground-state species. Vibrational coherence effects are also observable, and the motion of the wave packet is revealed by the change of the diffraction pattern with time, thus yielding the molecular dynamics. U E D provides the temporal evolution of the reaction coordinate directly and is wellsuited for studies of global structure changes on this time scale. Paper 5 details our experimental studies with UED and the current time resolution of the apparatus.

I. Introduction The interaction between a monoenergetic X-ray beam and the electronic distribution within a crystalline sample leads to the formation of a well-defined X-ray scattering pattern. The scattering pattern is highly sensitive to crystal structure; the inherent microscopic order of nuclei within the crystal, in terms of both spatial orientation and position, is projected into the macroscopic environment by the scattering process. Peter Debye proposed that structural information about gaseous samples would also be present in an X-ray diffraction pattern despite the lack of orientational and spatial order between mo1ecules.l The structural regularity of individual molecules is still present in a gaseous sample, and manifests itself in the form of concentric rings (as opposed to the spotlike patterns from crystals). The ring pattern is related to a probability distribution of internuclear distances and reflects the relative locationsof each atom, although the bond angles and the exact three-dimensional structure are not immediately evident. Gas-phase X-ray diffraction was first demonstrated in 1929, when Debye and co-workers imaged the scattering pattern of carbon tetrachloride vapor.2 The experiment was more challenging than crystallographic studies because of the low density of the gaseous sample, exposure times of several hours, and the vacuum requirements. Mark and Wierl discussed the feasibility of using electrons to study gas-phase molecular structure instead of X-rays. Electrons interact directly with the atomic nuclei such that electron scattering is several orders of magnitude stronger than X-ray scattering from molecules. Mark and Wierl were successful;in 1930 they produced a diffraction pattern from carbon tetrachloride that was more distinct than similar X-ray scattering exposures and required a fraction of the exposure time.3 The utility of gas-phase electron diffraction (GED) was recognized by several investigators,including Linus Pauling here at Caltech who was interested in using structures to examine the fundamental nature of the chemical bond. Currently there are National Science Foundation Pre-Doctoral Fellow. *Contribution No. 8916. .Abstract published in Aduance ACS Abstracts, February IS, t

1994.

0022-3654/94/2098-2766so4.5o/o

many GED laboratories around the world. The traditional experiment begins with a continuous, monoenergetic electron beam (10-100 keV), typically produced froma hot cathodesource. The electrons are collimated and directed toward a gaseous stream, where they scatter and are detected with a photographic plate. Analysis of the diffraction patterns, however, requires care, particularly when the molecule has many atoms. The advent of computer fitting techniques has greatly accelerated the analysis process.4 The structures of transient intermediates, such as radicals, was originally studied on the millisecond to microsecond time scale by flash photolysis combined with spectroscopicprobing.%' In the past decade, several groups have invoked electron diffraction as a method for examining these structures. Ischankoet al. created 1-ps electron pulses using an electromagneticchopper and studied the structure of radical products formed in IR multiphoton dissociation of CFJ8 (for a recent review see ref 9). Rood and Milledge conducted diffraction studies on the decompoeition of ClOz using 100-l.tselectron pulses,1oand recently Bartell and his group studied the phase changes of clusters on the microsecond time scale as welLL1 Ewbanb et ai. improved the temporal resolution of electron diffraction to nanoseconds by combining a laser-initiatedelectronsourcewitha linear diodeanaydetector,12 and their studies on the laser-decompositionof CS2 are supported by theoretical calculations. Mourou and Williamson demonstrated that the diffraction pattern from an aluminum film could be recorded with a single 100-ps electron pulse and that a phase transformation of the film could be observed in real time.') With this resolution, Elsayed-Ali et ai. have successfully shown reflection high-energy electron diffraction (RHEED) of laserheated surfaced4 (Kitriotis and Aoyagi recently reported nanosecond RHEED results's). Solution studies have also been considered; Bergsma et ai. theoretically predicted the ultrafast X-ray scattering patterns of solvated iodine; see section IV,part B.16 Structural changes of isolated molecular systems and reactions on the femtosecond-picosecond time scale have been probed with femtosecond transition-state spectroscopy (FTS), where the experimental time resolution allows observation of the coherent 0 1994 American Chemical Societv

Ultrafast Electron Diffraction nuclear motion.17 With this in mind, we proposed some time ago that GED could be used as a variant of the FTS approach to follow changes in internuclear separations with time for systems The premise is similar to FTZ: a laser in a molecular with femtosecond temporal resolution initiates a chemical reaction within a molecular beam sample, and a second pulse, in this case an ultrashort electron burst, probes the structural change.18 Two years ago we reported on our first efforts in this direction with an electron time resolution of -1 ps.19 These electrons (generated by the photoelectric effect) were accelerated to high energy (15-17 keV) and directed toward the interaction region, where the femtosecond pump laser, electron beam, and molecular beam intersect. The electrons were scattered by molecules in the interaction region, and the resulting diffraction pattern was detected with a two-dimensional charge-coupled device (CCD) operating in direct electron-'-?mbardment mode. The success of these experiments relied on minimizing the space-charge effects to maintain picosecond time resolution, and therefore the beam current was kept very low-up to 8 orders of magnitude below conventional GED beam currents. Tocompensate, single-electron detection was invoked. In conventional FTS experiments the time delay is varied by delaying the probe pulse relative to the initial t = 0 pulse in a Michelson interferometer arrangement. A similar design is used in UED, but the situation is more complicated because the electron and photon beams have different velocities. This velocity mismatch was recently shown to be important to the overall time resolution and must becarefullyaccounted for in theexperimental design.20 Considerations of temporal and spatial resolutions in our apparatus are further discussed in the accompanying paper.2' An important feature of ultrafast electron diffraction is the formation of a wave packet in the molecular sample, as occurs with FTS. Both the rotational and vibrational motions of the molecules are coherent; a degree of order is imposed on the previously isotropic sample. It is therefore necessary to consider thestructural changes, both in this regimeof coherent elementary dynamics and when coherence is subsequently lost. Conversely, it is possible to exploit the coherent motion to observe new structural features, as discussed below: In this article, we turn our attention to the impact of rotational and vibrational coherences on the scattering pattern in an ultrafast electron diffraction experiment. We wish to establish the nature of the structural and dynamical information that can be obtained from an experiment which combines high temporal resolution with high structural resolution. It is shown that an additional dimension of imaging can be achieved at times when coherence is induced or recovered-in a sense, Debye's ring pattern begins to approach the diffraction from a crystalline sample. We will specificallydiscuss the diffraction patterns which result from the temporal change of anisotropic distributions and show that more structural information, including bond angles and rotational constants, is available than in a conventional experiment. To predict the impact of vibrational coherence on structural changes, we present studies of the time-dependent scattering patterns for both dissociative and bound-state systems with the help of molecular dynamics simulations. We illustrate our points by first considering the structural changes in diatomics and then examining more complex molecular systems of, for example, five or eight atoms. The paper begins by highligh ting relevant aspects of gas-phase electron diffraction theory. This brief outline provides a basis for much of the theoretical analysis presented later, when we consider the effect of coherence on the diffraction pattern. Rotational coherence is discussed in section 111, and we show "snapshots" of the diffraction patterns that result from different combinations of polarization and excitation. We also examine the impact of rotational coherence on the analysis of dissodiation products and molecules with more structural complexity than a diatomic. The effects of vibrational coherence are considered in section IV. Our conclusions are presented in section V.

The Journal of Physical Chemistry, Vol. 98, No. I I, 1994 2767

II. Time-Independent CED Theory The general theory of gas-phase electron diffraction has been described elsewhere (see, for example, refs 4, 22, and 23). This section presents a theoretical framework with which to consider the effects of rotational and vibrational coherence. The incident electron beam is defined by wave vector b,where the magnitude of is related to the de Broglie electron wavelength by lbl = 2a/X. The incident electron beam is perturbed by the electronic potentials of the molecular sample and subsequently scatters with wave vector k at an angle 0. If the scattering process is elastic lkl), then the momentum transfer vector s may be written

sZ16-k

(1)

The magnitude of the momentum transfer is

Is1 = 21161 sin(6/2) =

4a

sin(8/2)

(2)

The total scattering intensity, I, is the sum of contributions from atomic (I*) and molecular scattering (fM). These intensities vary with scattering angle and are usually expressed as a function of s rather than B

If it is assumed that the electronic potentials of each atom in the molecule are independent (the independent atom model), then the atomic scattering intensity may be written as a sum of elastic and inelastic scattering contributions

In this equation, N i s the number of atoms in the molecule,fi and

SIare the direct elastic and inelastic scattering amplitudes for atom i, a0 is the Bohr radius, and Cis a proportionality constant. The contributions from spin-flip scattering amplitudes (gi)have not been included as they are generally neglected for high-energy electron diffraction experiments24 The molecular scattering intensity is composed of interference terms which contain structural information about the scattering center. The intensity is written as a sum over all internuclear separations: N

N

7#I

where qris the complex phase term for the corresponding scattering amplitude fr, and rij is the position vector between atom i and atomj. The interaction between incident electron and scattering molecule is extremely f a s t - o n the order of attoseconds-and the molecular scattering intensity must be averaged over all populated vibrational and rotational states in the molecule. Vibrational averaging contributes an additional exponential term (the Debye-Waller factor) that is dependent upon I , the mean amplitude of vibration

This factor represents the replacement of a specific internuclear separation with a weighted distribution corresponding to the vibrational motion between the two atoms. If the molecules have not been prepared in a specific rotational state, then the rotational average is equivalent to an average over

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The Journal of Physical Chemistry, Vol. 98, No. 11, 199'4

all spatial orientations at the inst--it of scattering. The molecular scattering intensity for a spatially isotropic sample takes the form

wherejo is the zero order spherical Bessel function, or sin(sr,)/ srijThe scattering amplitudes,ft andfi, decrease with the square of s (ref 25), so the total scattering intensity decays like d;as a result, a graph of the modified molecular scattering intensity, sM(s),is better suited for revealing the fine details of the scattering signal than a plot of the total intensity

where a and b correspond to two atoms in the molecule. As mentioned above, the molecular scattering function contains all of the structural informationabout the molecule, but an illustrative interpretation of experimental results is provided by taking the sine transformof sM(s) andexaminingflr), the radial distribution function

f i r ) = JomsM(s)sin(sr) ds

(9)

The radial distribution function plots the relative density of internucleardistancesin the molecule. This transform relationship is strictly true only in the case of pure elastic scattering from a homonuclear diatomic molecule,23but f(r) is nevertheless very useful for a qualitative understanding of the structure of more complicated systems. The range of experimental scattering (e.g., 10-15 A-1 intensity is limited by some maximum value smax for the UED experiments reported in the accompanying paper), and the experimental radial distributionfunction typically includes an exponentialdamping term to filter out artificial high frequency oscillations f(r) = JosmusM(s)exp(-k,.,s2)

sin(sr) d s

(10)

It should be noted that all data analysis and fitting procedures are performed on the experimental molecular scattering function, and not the radial distribution. The scattering factors which appear in the above equations cf, 7,S ) are functions of the atomic number 2 of the scattering atom, the energy of the incident electron beam, and s. These factors may be obtained from published tables.25 All scattering intensities and molecular dynamics calculations reported here were computed with the C programming language on an IBM PC compatible. The incident electron energy was assumed to be 40 keV, ,s was limited to 15 A-1, and the radial distribution function damping constant (kd)was set to 0.01 A2. 111. Coherent Spatial Orientation

In UED experiments, the molecular spatial-orientation distribution is dependent on the polarization of the laser and the type of transition accessed. If Q is the angle between the molecular transition moment and the polarization of a linearly-polarized laser pulse, then immediately after a parallel dipole transition the number of excited molecules at angle Q is proportional to cos2 Q. If the excitation probability is unity, then one-third of the sample is pumped into a higher state and the remaining unexcited molecules must have a sin2 Q distribution. In effect, the pump laser converts the isotropic sample into an anisotropic mixture of excited and unexcited molecules with well-defined spatial orientation distributions. The duration of the anisotropy, however, lasts only until rotational motion randomizes the spatial orientation

Williamson and Zewail again: after several picoseconds the sample will again be isotropic, albeit a mixture of excited and unexcited molecules. Rotational motion is quantized and the rotational frequencies are commensurable (or nearly so), and barring any collisions, there will be some later time in which all the molecules return to their original spatial orientation. The short-lived anisotropic distribution will therefore appear again and again aver the course of many nanoseconds.26 This phenomenon, known as rotational recurrence, has been extensively studied in the context of timeresolved spectroscopy experiments, where rotational recurrences have proven useful for determining structures of large molec~les?~ The period of the recurrence in a symmetric top molecule is equal to 1/ ( 2 8 ) where 8,the rotational constant, is related to the speed of light c, Planck's constant h, and the moment of inertia 16 by B = h/8r2Zbc. For asymmetric tops, the key observables are A, B, and C. It has been shown that the temporal width At of the recurrence is related to the temperature T and the rotational constant by27

where B has units of cm-l and T i s in Kelvin. For example, the first recurrence of tram-stilbene (B = 0.252 GHz), occurring 1.95 ns after excitation, has a full width at half maximum (fwhm) of 21.Ops at 5 K and 6.8 ps at 50 K, and these calculations26have been experimentally tested. In a diffraction study of rotational recurrences, only spectroscopic information about the initial excitation transition is required; the recurrences would be probed by following the diffraction pattern in time, as opposed to further excitation to another state with a second ultrafast laser pulse. Kohl and Shipsey showed theoretically that the diffraction pattern from molecules in a specific rotational state, with a welldefined nutation axis, is not cylindrically symmetric and contains extra clues about themolecular structure.28 The molecular sample in UED, however, gains a different type of rotational order. Rotational states are coherently populated by the excitation laser, but the judicious combination of a dipole transition, quantized rotations, and high temporal resolution creates an ordered spatial orientation distribution. We might expect that the diffraction pattern of a spatially anisotropic sample will contain more structural information than a conventional GED pattern. Equation 5 must therefore be reexamined to ascertain what the impact of an anisotropic spatial orientation distribution formed by ultrafast laser excitation will be on diffracting electrons. These effects must be considered at short times, immediately after excitation, and over longer times, when the recurrences take place and the anisotropic distribution reappears. To emphasize the changes in the diffraction patterns, it will be assumed that the excitation probability is unity. Experimentally, however, the probability will be one-half at best because of stimulated emission and saturation effects, unless the upper state is dissociative or Rabi cycling is introduced. Fink et af., treated the alignment problem for molecules spatially oriented in a hexapole field.29 They have suggested that changes in electron scattering would be potentially useful for diagnosing the degree of spatial alignment in molecular beams, and their most recent experimental work has shown agreement with the theoretical predictions.30 The geometrical framework presented by Fink et af. describes their alignment studies well, but more degrees of freedom are necessary for modeling the experimental arrangement discussed here. We outline our methodology in part A of this section. Part B follows with a brief synposis of the diffraction patterns generated by molecules in total spatial alignment; this synposis will assist in interpreting the more complicated patterns from laser-excited samples. The theoretical diffraction equations for the molecular subset selected by linearly-polarized lasers in parallel and perpendicular configurations are established in parts C and D, and we will discuss their application to UED rotational recurrence studies of molecular iodine in part E. Part F concludes this section by

The Journal of Physical Chemistry, Vol. 98, No. I I . 1994 1769

Ultrafast Electron Diffraction

Fij(s) = (eiar'9Spa,ial

(13)

Fjjcontains the spatial average of the scattering function for a specificinternuclear separation. Note thatFjjisa functionofthe momentum transfer vector s, and not just its magnitude. Figure 1 presents the geometrical frame of reference we have chosen to evaluate Fjp The incident electron beam travels along the z-axis, and the position vector rjj is oriented relative to the z- and x-axes by an altitude angle Q and an azimuthal angle $, respectively. Diffracted electrons strike the two-dimensional detector at a position determined by the scattering angle 8 and the rotational angle @ (defined relative to the x-axis)." In this coordinate system, the momentum transfer vector is written:

Scattering Center

Figure 1. Electron scatteringgeometrydefined in the text. The incident electron beam travels toward the scattering center along the z-axiswith wavevector h.The position vector between atoms i and j , ru. is defined byaltitudeanglen(relativeto ther-axis) andazimuthalangle+((relative to thex-axis). Electronsscatter at angle0 with wavevector k and intersect theplaneofthedetectorat angle$ (relativetothex-axis). Themomentum transfer vector is s. In a UED experiment, the excitation laser travels toward the scattering Center along the y-axis.

s = k,(-cos

@ sin 8, -sin @ sin 8, 1 -cos

8)

= s(-cos $ cos(8/2), *in @ cos(8/2), sin(8/2))

(14)

and the position vector is written

rij = rij(sin Q cos $, sin Q sin $, cos 0 )

(15)

The dot product of these two vectors simplifies to

siij = srij[sin(8/2) cos Q - cos(8/2) sin Q cos(@- $)I

Y

Todetermine Fil,thescattering function ei'"umnst beintegrated over thesurfaceof thespheredefined by nand$. Theanisotropic spatial orientation is included by multiplying the scattering function with an appropriatedistribntion function,P(Q,$), before integration

"8

b, 8s:

0G."3

(16)

ec-

-#la

Molecular Sample (Isotropic)

Detector

FigureL Electronscatteringfromanisotropcsampleofmoleculariodine. On theieftisaschematicoftheinterscnionbetween theincidentelectrons and the molecular sample. On the right is a perspective view of the resulting modified molecular scattering pattern, sM(s). The scattering pattern consists of cylindrical oscillations with frequency r1.112~.

evaluating the structural information present in the anisotropic diffraction pattern from a more complicated molecule, trifluornmethvl inrlid~.

A. Theoretical Framework. The scattering intensity, I,, of a laser-excited molecular sample must be extracted from eq 5 , and to simplify this derivation we will initially consider only the scattering from position vectors rij which are parallel to the molecular dipole transition axis. This is sufficient for describing all diatomics and linear molecules; more complicated molecules with nonparallel position vectors will be considered in part F. The vibrational period of most chemical bonds is significantly faster than a few picoseconds, and thus the starting point for this analysis is actually eq 6, where the vibrational motion has been separated from the average. In the context of UED, the average over rotational states should be evaluated as an average over spatial orientation since many rotational states are populated; the important perturbation is the initial (and subsequent recurrence) spatialorientationdistributionselected with theexcitation laser polarization. Equation 6 may he rewritten as:

cccKjKl N

Ids) =

N

exp(- ~i~sz)Re[ei'"'"JFij(s)l

i=1 j=1. I*J

where

(I2)

Mathematicaldetailson theevaluationofthis integral for several relevant distribution functions have been consigned to the Appendix. For the case of an isotropic distribution (P(Q,$) = I), the results is:

asexpected. andthccorresponding two-dimensional plot ofsM(s) for molecular iodinc is presented in Figure 2 uith a drawing of the molecular sample. The frequency of the oscillations in this familiar ring pattern is equal to q.t/2x. wherc rl.l = 2.666A.I2 B. Diffraction from Total Alignment. With the theoretical framework defined in part A, electron diffraction patterns from a laser-prcparcd molccularsamplc may be predicted. It is useful. howevcr,tofirstconsider thescattering pattern froma molecular sample with tntal spatial alignment. The functional forms of there pattcrnr require no integration and may bederivcd dircctl) from cqs 6and 16. Twodifferent fundamental arrangcmcntsare possible: (1 J all molecules arc oriented parallel to the direction of the incident electron beam, and ( 2 ) all molecules are oriented perpendicular to the direction of the electron beam. In the firrt case (0= 0, thc bond axis is parallel to the z-axis). the Scattering function for iodinc is equal lo

The diffraction pattern is cylindrically symmetric and its section is an oscillatory function that changes slowly with s at small scattering angles (ka >> s). As shown in Figure 3, the modified molecular scattering pattern is an order of magnitude more intense than electron scattering from an isotropic distribution, but the slow onset of the oscillations makes it difficult to extract structural information. This has a physical interpretation-the molecules are struck by the electrons end on, and the second atom is "hidden" by the first atom along ko. Although

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The Journal ofphysical Chemistry. Vol. 98, No. 11, 1994

Y

* 9

8'

Si8

88%

Molecular Sample (Perpendicular) Iletector

s (A-I) Figure 3. Modified molecular scattering cross sections from diatomic iodine in three different spatial distributions: (1) isotropic, (2) total alignment parallel to the incident electron beam (z-axis), and (3) total alignment perpendicular to the incident electron beam and parallel to the x-axis. The scattering intensity from the isotropic sample is an order of magnitude lessthan thescattering intensity from spatially alignedsamples. The oscillation frequency of interference fringes from the parallel configuration is very slow for small s; the first cycle finishes at s = 22.2 A-'. Scatteringfrom the perpendicular configurationis not cylindrically symmetric. The interference fringes along the x-axis (parallel to the internuclearaxis) arestrong,and the frequency issimilar tothe frequency oftheisotropicfringes. Therearenooscillationsalong they-axis (normal to the internuclear axis). the number of cycles increases with the square of s, there is little information present in the range of s available to UED. In the second case, when all molecules are spatially oriented perpendicular to the electron beam (R = 712, J. = 0, the bond axis is parallel to the x-axis), the scattering function is

Theprescnccofthccosotermdestroys thecylindricdlsymmetry. Thcreare nooscillationsa1ongthe)-axis, whereo = f ~2 (Figure / 3). while the oscillations which exist along the x-axis have a period similar LO the intcrlercncc fringes from an isotropic distribution and are an order of magnitude larger. A tuo-dimensional plot of s M ( s ) for this second case. shown in Figure 4. clarifies the connection hctwecn the horizontal and vertical cross sections presented in Figure 3. The diffraction pattcrn from spatially-oriented molecules parallcl to the x-axis is a series of linear fringes, as opposed to cunccntric rings. This figure is similar tothe interference pattcrn from Young's doublc. in fact. the scattered electron slit light bcattcrine- exocriment; . intensity of eq 20 may he wriken in a'form analogous to the diffractedlight intensity froma pairofslitsseparated hya distance d3'

(for 6 = 0) (21)

The potential wells of the two iodine atoms effectively function as a pair of "slits" for the incoming electrons, and the pattern

..

Figure 4. Electron scattering from a sample of molecular iodine with total spatial alignment. The bond axis ofeach molecule is parallel to the x-axis. On the left is a schematic of the intersection betwecn the incident electrons and the molecular sample. On the right is a perspective view afthecorrespandingmodifiedmolec"larscattering,sM(s).Theamplitude ofthe interference fringes is approximately an order of magnitude larger than the interferencefringes from the isotropic sample (see Figures 2 and 3).

shown in Figure4resultsfrom interference between theelectrons diffracted by these two scattering centers. Total alignment of the iodine molecules has affected both the amplitude and the frequency of the interference fringes in the diffraction pattern. The amplitude has increased by an order of magnitude, which implies that the signal-to-noise ratio for molecular scattering from spatially-oriented molecules is larger than isotropic scattering. We might expected that this increase in signal-to-noise will counterbalance the decrease in the percentage of molecules contributing to the anisotropic diffraction pattern in an actual experiment. The oscillation frequency of the diffraction signal from the first case (the parallel configuration) is significantly different from the s a n d case (the perpendicular configuration). Structural information is more difficult to obtain from a parallel configuration because the first few oscillations are very slow, and the diffraction pattern would have to be measured out to large values of s. The oscillation frequency of the fringes in the perpendicular configuration, rt.I( 1 - (s/Zk0)911'/27, approaches the frequency of the isotropic distribution,r1_1/2~,athigh electron energies where ko is large. As will be emphasized in part F, the perpendicular configuration also contains information about bond angles. Thediffraction pattern in Figure4 tells theexperimentalist that the iodine bond axis is parallel to the x-axis; if J. is nonzero, then the interference fringes would be similarly rotated on the detector hyangleJ.relative to thex-axis. Thediffraction pattern from more complicated molecules is a sum of contributions from individual atom pairs, and interference fringes in the diffraction pattern will reflect the differing spatial directions of the various rjj. C. Parallel Excitation. The laser beam, electron beam, and molecular beam are all mutually orthogonal in our UED e~periment.~ This ~ , ~experimental ~ design is motivated by ease of construction and a desire to reduce velocity mismatch broadening in the total temporal resolution of the experiment.z0 We will now consider the diffraction patterns which result from molecules excited by a linearly polarized laser traveling toward the scattering center along the y-axis, such that the polarization vector is parallel to the wave vector b. For now it is assumed that scattering takes place before the molecules have time to rotate out of their initial distribution and that the entire subset of molecules is excited. The scattering pattern at later times will be considered in part E. The spatial orientation distributions of both excited and unexcited molecules will be cylindrically symmetric about b, and the diffraction pattern should reflect this symmetry. For a

The Journal of Physical Chemisrry. Val. 98, No. 11, 1994 2111

Ultrafast Electron Diffraction I

~

I*otroPic

Excited

---- Unexcited

1

X l L

-10

2

s,

Y

Molecular Sample (Perpendicular 1 Excited) Figure 5. Modified molecular scattering c r w scctioni of ground-rmtc i d i n c from spatial urientation di,tribuuonscrcated by a linrirly pulanied laser. The polariration vector is parillel to ko and the molcculu dipole trmsitiun i s a w m e d to be parallcl.sothc I~WCIEIIC muleculcsurwntcd porollpl to the electron beam Thc intcnrit) of the in1crfcren:c fringes from thi,subsct ofthc molecular sample isan ,irdcruimagnitLdr.rmaller than the interfcrcncc fringcs from the isotropic simple. The 0,cillition f r c q x n c y is similar. but r 2 ridianr out ut phdrc. 'The intcrfcrencc fringes from the unexcited sub,et are m x l a r i n inlcnsit) and frcqucnc'! to

the isotropic m t i r r m g pattern

parallel dipole tranbition. theorientation of the intcrnuclcar axis ofthecxcitcd molecules will haveaspatialdisrributionofP(R.J) =cosZR Thcspaual di5tribution ofthe unexcited molcculcsuill therefore be P(R.$)=sinzl.l. Turciter~te,ueareonlyconsidering position vectors which 3rc parallel to the cxsitation axis, and the excitation probability IS unity. Integration of eq 17 using these distribution5 yields the following functions for F,, (see Appendix) parallel, excited

parallel/unexcited

.-

Detector Figure6. Electron scattering from thesubsetofa ground-statemolecular iodine sample excited by a linearly polarized laser. The polarization vector is parallel to the x-axis and the molecular dipole transition is assumed to be parallel. On the left is a schematic of the intersection between the incident electrons and the perpendicular/ercited subset of the molecular sample. On the right is a perspective view of the corresponding modified molecular scattering, sM(s). The scattering pattern resembles a cross between diffraction from an isotropic sample (Figure 2) and a totally aligned sample (Figure 4). The interference fringes along the x-axis are similar in intensity and amplitude to the interferencefringes from an isotropic sample. The interference fringes along the y-axis are weak and resemble the scattering pattern from the parallel/excited subset shown in Figure 5 .

D. Perpendicular Excitation. Significantly different effects are observed when the polarization vector of the laser is parallel to the x-axis and perpendicular to ko. As in the example of total perpendicular alignment, the scattering pattern loses its cylindrical symmetry because the spatial orientation distribution selected with the polarized laser is not symmetric about the z-axis. With an excitation probability of unity, the distribution of the excited moleculesisdefinedby P(Q,$)=sin2acosz$andthedistribution of the unexcited molecules is (1 - sin2 a cos2 $). Integration of eq 17 with these distributions is discussed in the Appendix and leads to the following functional forms for Fij: perpendicularJexcited

j, b v 1

Fii(s) = jo[sr..] -~srii 'J

+ sin2(O/2)j2[srii]

(24)

As expected, these functions are independent of 4. The modified molecular scattering function ofthese two distributions for groundstate iodine are plotted with the isotropic distribution in Figure 5 . The scattering contribution from the parallel/excited subset of molecules is relatively weak; evidently the strong scattering signal observed in part B (when all molecules are oriented parallel to the electron beam) is rapidly washed out by interference with scattering from molecules slightly off-axis to the electron beam. Because this contribution is weak, the oscillation frequency of thescattering intensity from the parallelJunexcited subset isnearly identical to the isotropic distribution. The oscillation frequency from the parallel/excited subset is similar, but phase-shifted by ~ 7 1 2 Sincealldistributionsarecalculatedfor . thesamespecies, the sum of the two scattering signals is equal to the diffraction signal from the isotropic distribution. Consequently, if the excited species is structurally identical to the unexcited species, the diffraction pattern will not change. Structural information inthe diffraction pattern from a sample with total parallel alignment was identical to the information from an isotropic distribution but more difficult to extract, and thesameis trueforthediffractionsignal from theexcitedJparallel subset. Like the isotropic pattern, the excitedJparalle1 signal depends only on the magnitudes of the position vectors. It would bedifficult, however, todetermine the molecular structureof the excited-state species because the low amplitude signal is masked by the diffraction signal from the unexcited molecules.

+

Fij(S) = j'[sriil cos2(S/2)j2[srii] cos26 sr..'I ~

(25)

perpendicularJunexcited

As in the parallel case, the sum of these two functions is jo[sr,]. the iso:ropic scattering distribution, but note that the presence of the cos2 6 term means that each function is not cylindrically symmetric. A two-dimensional plot of sM(s) for the perpendicular/excited subset of molecules is shown in Figures 6 and 7a, and theperpendicular/unexcitedcaseisshowninFigure7b.These are calculated for ground-state iodine, Both patterns appear to be a cross between isotropic scattering (Figure 2) and scattering from a sample with total perpendicular alignment (Figure 4), and despite the fact that only one-third of the molecules are in the perpendicular/excited subset, the amplitudeof the interference fringes along the x-axis is as strong as the isotropic distribution. Furthermore, the 4 dependence of the fringes indicates the directionality of r,., as was discussed in part B. The two-dimensional radial distribution functions of the isotropic distribution and the perpendicular/excited distribution are shown in Figure 7c,d. Admittedly, taking the sine transform of the function in eq 25 to extract the radial distribution is a crude approximation, and this is reflected in the fluctuating baseline of the surface (Figure 7d). We will consider more

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The Journal ofphysical Chemistry, Vol. 98, No. 11, 1994

Internuclear Separation

(A)

Internuclear Separation

(A)

Figure 1. Diffraction from ground-state iodine. (a) sM(s) for the perpendicular/excited subset, This pattern was also shown in Figure 6. (b) sM(s) for theperpendicular/unexcited subset. Theunexcited molecules are aligned parallel to the excitation laser (along they-axis), sotheinterference fringes are strong along the y-axis and weak along the x-axis. (c) Two-dimensional radial distribution curye from an isotropic sample of ground-state iodine. The ring is located at -2.1 A. Only one quadrant is shown. (d) Two-dimensional radial distribution curve from the perpendicular/excited subset of the molecular sample. The intensity at 2.1 A is strong along the x-axis and weak along they-axis. This reflects the physical orientation of the iodine sample in the labratory frame of reference. Only one quadrant is shown. appropriate techniques for extractingf(r) in future publications. For now, however, the sine transform is sufficient to provide excellent qualitative information, and when scattering from trifluoromethyl iodide is considered in part F, the radial distribution function will provide a description of the structure that is easier to interpret by visual inspection than the molecular scattering function. As might be expected from the appearance of the perpendicular/excited molecular scattering function, the corresponding radial distribution function also indicates the directionality of the iodine-iodine internuclear separation during the experiment. The disappearance of the 2.7-A peak along the y-axis reflects the fact that most molecules oriented parallel to the laser beam axis are not in the perpendicular/excited subset aftera parallel dipole transition. The radial distribution function corresponding to the perpendicularlunexcitedsubset(not shown) contains a peak along the y-axis and no peak along the x-axis. E. Experimental Applications: Ground- and Excited-State Structures. Thediffraction pattern from laser-prepared molecules differs from the isotropic pattern because excitation is a selective process that depends on the spatial orientation of the molecular dipole. Thisdifferencewillnot beseen, however, ifthemolecular structuresofthe twostatesaresimilar becausethetotaldiffraction pattern is always a sum of scattering from the excited and the unexcited subsets; if the two species have identical r,, then this sum is just jo[srjj]. Fortunately, the excited-state structure is usually changed from the ground-state species to some degree,

and electron diffraction is quite sensitive to small changes in molecular structure. For example, Figure 8 shows that there is a clear difference between the radial distribution curves corresponding to the lowest vibrational levels of ground-state iodine andof B-stateiodine. In thenext few paragraphs, we willconsider how to combine rotational coherence effects with the diffraction patterns discussed in parts C and D such that more structural information about thescattering molecules can be extracted than in a conventional gas-phase electron diffraction experiment. With an isotropic distribution, it is possible to improve the signal-to-noise ratio by summing the total scattering intensity a t a constantvalueofson thedetector. When the laser polarization is parallel to the electron beam, the diffraction patterns from excited and unexcited subsets are cylindrically symmetric and may therefore beaveraged ina similar way over theentiredetector. We haveseen, however, thatthisarrangementonly leads tosubtle changes in the diffraction pattern, and that these patterns do not appear to contain any extra structural information. We will instead turn our full attention to excitation with the laser polarization perpendicular to ko. In this case, the diffraction pattern loses cylindrical symmetry and the signal cannot be averaged over the entire detector. Variations in thediffraction pattern must be particularlydistinct, and ideally there would be some way to separate the scattering signal of the excited species from the unexcited species. The scattering patternat early times,in theinstant after themolecular

-b

The Journal of Physical Chemistry, Val. 98, No. 11. 1994 2113

Ultrafast Electron Diffraction Ground

Exclled State

s

although they will have undergone several vibrations. Perhaps the simplest experiment of this type is excitation of iodine from the X state to the bound B state (a parallel dipole transition). For demonstration purposes, the influenceof Franck-Condon overlap will be ignored, and it is assumed that the excitation laser places ground-state molecules into the lowest vibrational level of the B state. The ground state molecules are also assumed to be in the lowest vibrational level. (A more detailed investigation of the B X transition must account for the impact of overlap and vibrational motion and is delayed until section IV.) The resulting diffraction pattern is a sum of the perpendicular/excited distribution (eq 25) evaluated for the B state (11.1 = 3.0267 A, 11.1 = 0.04598 A) plus the perpendicular/unexcited distribution (eq 26) evaluated for the X state (11-1 = 2.6664 A, 11.1 = 0.03519 .&).I4 The modified molecular scattering pattern from this system (Figure 9a) reflects the anisotropy of the sample. It is clear that the oscillation frequencies along the x- and y-axes are different, but extracting informationon the excited-statespecies is nontrivial because the background signal from the unexcited molecules also depends on 4. Other possible approaches should be considered, as discussed below. Iodinemay beexcited toadissociativestate insteadofa bound state. The dissociation time is very fast-within a few hundred femtoseconds"-so shortly after excitation the molecular sample will consist of atomic iodine in the perpendicular/excited distribution and molecular iodine in the perpendicular/unexcited distribution. The diffraction pattern will therefore he equivalent to Figure 7b. The change in the diffraction pattern is certainly distinct, and additional structural information about the ground state species has been determined-the directionality of rl.l. Over longer times, the anisotropicdistribution ofthe unexcited iodine molecules will be lost due to rotational motion, and the electron diffraction pattern will correspond to an isotropic distribution. The amplitude oftherings will be two-thirdsofthe early-time amplitudealong thex-axis,assuming that the excitation laser has dissociated all of the molecules in the perpendicular/ excited subset. Approximately 1 / 2 8 picoseconds after laser excitation, however, a rotational recurrence will take place and the anisotropic distribution will appear once again. (A halfrecurrence will also occur earlier at 1/48 P S . ) ~By ~ taking a series of diffraction patterns in time, it will be possible to mimic the spectroscopic rotational recurrence experiments of Felker et ai. and determine the rotational constant with UED. The experimentalist looks for those times at which the isotropic distribution changes to the anisotropic pattern. Rotational recurrencestudiesoftheground-state species are therefore possible with UED. Observing rotational recurrences with UED is a time-domain technique for measuringrotational constants, but these rotational recurrences are useful for distinguishing bctween the structures of ground-state and excited-state species. Recall that the earlytime diffraction pattern from excitation of the B X transition was anisotropic hut did not distinguish well between the excitedand ground-state species. This will not be the case at later times because the rotational constants of the ground- and excited-state species are different, and the recurrences will be separate and distinct. As an example, with ground-state iodine (v = 0). EX = 0.0373 cm-I and 1/2& = 447 ps; for iodine in the B state ( v = 0), BE = 0.0289 cm-' and 1/2& = 577 ps." These recurrences are well-resolved in time, and the scattering pattern at each recurrence will correspond to a sum of one species in an isotropic distribution and the second species in its initial anisotropic distribution. Studies of the excited-state species are best conductedwithanunpolarizedlaser,~~ thatthediffractionsignal for the excited species is a sum of the parallel and perpendicular distributions. Although the parallel contribution does not add much to the scattering pattern, the isotropic background signal from the unexcited molecules is reduced by a factor of 2. The excited pattern is easier to discern because only onc-third of the sample remains unexcited.

-

0

1

2

3

4

5

Internuclear Separation (A) Figure 8. Radial distribution curves for ground-state (X) and excitedstate (B) iodine at vibrational level Y = 0.

0 5

5

10

10

-

0 5

5

10 10

Figure9. Modified molccularscatteringcuwesfromamiitureofgroundstate and B-state iodine. (a) Scattering pattern immediately after excitation. The excited iodine is in a perpendicular/excited distribution and the ground-stateiodine 1s in a perpendicular/unexciteddistribution. The oscillation frequency along the x-axis is slightly greater than along they-axis, which reflects that B-state molecules (q.1 = 3.027 A) in the sample are oriented parallel to the x-axis and the unexcited molecules @ . I = 2.666A) areoriented parallel to thcy-anis. (b) Scattering pattern a few picoseconds after excitation. The spatial anisotropy has been lost due to rotational motion. The diffraction pattern now corresponds to an isotropic sample of'/, ground-state icdine and I / , B-state icdine.

sample has been excited by the polarized laser, should be considered first. The excited and unexcited molecules have not hada chanceto rotateout oftheiranisotropicspatial distributions,

Williamson and Zewail

1114 The Journal of Physical Chemistry. Vol. 98, No. 1 1 , 1994

Figure 11. (a) Structure of trifluoromethyl iodide. (b) Spatial distributions of the four CFlI pition vectors after a parallel molecular dipole IransitionalongtheC-Ibondaris has beenexcited. The threcremaining position vectors (TCF. rF.+. rF-1) sweep out cones definedby an angle Rtj. where Rjj is the angle between IGI and rp

0

s,

5

10

10

which were parallel to the molecular dipole transition axis. To correct this, we willexamine thescattering pattern fromall position vectors in CFJ after the moleculehas been excited along a dipole transition parallel to the C-I bond. The structure of CFJ is shown in Figure 1la. Excitation of a dipole transition parallel to the C-I bond is insensitive to the angular position of the fluorine atoms around the C-I axis, and thescatteringpattern must amount for an averageoverall possible rotational orientations of the other internuclear separations. As demonstrated in Figure 1Ih, each position vector r, sweeps out a cone about the C-I axis; the angle between the C-I axis and each cone is defined by RIP The geometry of this arrangement is shown in Figure 12. The angle a defines the angular position of rv about the excitation axis, and in the primed coordinate system, rlj is

rij = rij(sin Rij cos a,sin Rlj sin a,0) 0

s,

5

.

'

10

(27)

where Olj is the angle between rij and the excitation axis. The primed m r d i n a t e system may he transformed to the unprimed coordinate system according to

10

Figure 10. Modifiedmolecularscatteringcurvesfromamixtureofground-

state and B-state iodine during subsequent rotational recurrences. (a) 447 ps, ground-state rotational recurrence. The molecular sample has been excited witha linearly polarized laser polarization parallell ox-axis)

and is ' 1 3 B state, 2/3 ground state. Ground-state iodine returns to a perpendicular/unexcited spatial distribution. B-state iodine remains isotropic. (b) 577 ps. B-staterotationalrecurrence. The molecularsample has been excited with an unpolarized laser and is therefore B state, ' 1 3 ground state. B-state iodinereturns to an unpolarized/excitedspatial distribution. Ground-state iodine remains isotropic. The scattering patterns for molecular iodine a t 447 and 577 psarepresented in Figure IO. At 447 ps, theinterference fringes corresponding to ground-siaie iodine are enhanced along the y-axis; similarly, a t 517 ps the interference fringes of exciiedSioie iodine are enhanced along the x-axis. In both figures, the x- and y-axis scattering profiles are different in frequency and amplitude; separation of the ground- and excited-state scattering patterns has been accomplished by exploiting the differences in the period of the rotational recurrences in UED. Note that the temporal and spatial characteristics of this experiment enhance the capability of electron diffraction to give both ground-state and excited-state structures. F. More Complicated Molecules, CF& The enhancement of the interference fringes shown in Figure 10 is perhaps a little misleading because iodine is a simple diatomic molecule. In general, the scattering pattern of a molecule with many atoms is a sum of different frequency components from the various internuclear separations, and therefore the total signal results from a combination of constructive and destructive interference between these different components. Furthermore, the scattering theory developed in part A does not adequately describe more complicated molecules because we only considered position vectors

sin R cos $ = rlj cos Ro sin R sin $ cos0 cos ! cos l $ -sin $ sin R cos J. cos R sin $ cos $ sin R sin $ +in O 0 cos Application of this transform to eq 27 leads to

El

1

[

[

+

][I:]

(28)

r.. =

(sin R4 cos a ws D + cos Do sin D) cos *-sin Oil sin a sin J. rlj (sin O,j COS a cos R ws R, sin R) sin *-sin D,j sin a cos J. cos R4 cos R - sin R, cos 01 sin J.

[

+

1

(2% Note that if rlj is parallel to the excitation axis (O,j = 0)then eq 29simplifiestotheearlierdefinitionofr4(eq15). Themomentum transfer vector, s, is not dependent on R, and a,but Fij must now include a third integration about the circle defined by a and r, sin 0,

It is important to note that the weighting distribution P is independent of Rlj because P depends only on the angle between the excitation axis (the C-I bond in the example discussed here) and the laser polarization axis. The inner integral may be evaluated analytically using an identity discussed in the Appendix. The resulting double integral was integrated numerically using

Ultrafast Electron Diffraction

The Journal of Physical Chemistry, Vol. 98, No. 11, 1994 2775

Mathematica and analysis of these molecular scattering curves led to the following equation for the perpendicular/excited distribution when Qij is arbitrary

This equation agrees with the numerical integration results to at least ten decimal places; note that eq 31 reduces to eq 25 when

n, = 0.

The structural constants rfj, 111,and nufor CF31are summarized in Table 1.36 The two-dimensional radial distribution curves corresponding to the perpendicular/unexcited subset of CFJ are presented in Figure 13. The isotropic pattern consists of three rings: the inner ring corresponds to the C-F atom pairs, the middle ring is an overlap of the C-I and the F-F atom pairs, and the outer ring corresponds to the Fa-I atom pairs. A cross section of this cylindrically symmetric pattern is shown in Figure 14. All three rings are present in the pcrpendicular/unexcited distribution, but now the amplitude of each ring varies with the angle 4. The order in the molecular sample (imposed by the anisotropic distribution) provides more structural information in the diffraction pattern by separating the atom pair components as a function of 4. The amplitude of the middle ring is approximately half as large as the isotropic distribution because the C-I and the Fa-F components are orthogonal. The C-I scattering contribution should be similar to the molecular iodine pattern in Figure 7b; since rc-I I rF-F (QF-F = goo), the contribution from F.-F atom pairs is closer to the pattern in Figure 7a. The sum of the two contributions leads to a complete ring at -2.15 A. The area of a radial distribution curve peakis proportional to n&Zj/rij, where nij is the number of identical atom pairs present in the molecule and Zi b the atomic number of atom i. In many cases it would be possible to idenfity the two components of the middle ring because their maximum peak areas would be different, but in this 328.It is still relatively simple example it happens that 2~21to identify the components, however, because the C-I interference fringes are damped by the cos(qc - 71) phase term and the F-F interference fringes are not damped. Consequently, the F-.F peak is expected to be narrower and taller than the C-I peak, and this is confirmed in the cross sections (Figure 14). The perpendicular/unexcited radial distributions in Figure 13 and in the profiles of Figure 14 provide qualitative information about the angles QC-F and &...I. A quick glance at the inner ring shows that the C-F bond is almost orthogonal to the C-I bond because the peak is larger on the x-axis than on the y-axis. Conversely, the amplitude distribution of the outer ring shows that Q F - ~ must be closer to Oo (or 180O) than 90°. It will be possible to determine aij with great precision by carefully fitting thex- and y-axis molecular scattering curves, and a more detailed discussion of this process will be presented in future publications. In X-ray crystallography, the entire structure of the unit cell is mapped out by taking a series of diffraction patterns using different crystal orientations. This process can be imitated in UED to some extent by exciting different dipole transitions-the perpendicular/unexcited radial distribution of CF31 will be different if the excitation axis is parallel to a C-F bond. These diffraction patterns thus contain information about the nature of the excited transition (perpendicular versus parallel, etc.), in addition to structural details. Finally, rotational recurrence UED provides a unique opportunity for determining the rotational constants and structure of radicals and other dissociation products. If the dissociation takes place more quickly than the duration of the rotational recurrence, then the dissociation products will have anisotropic spatial orientation distributions. For example, CF31readily dissociates into CF3 and atomic iodine when excited by a 266-nm laser pulse.

Figure 12. Scattering geometry for a position vector ri, that is not parallel to the excitationaxis. The excitation axis (2’) is defined by altitude angle Q and azimuthal angle tj (see Figure 1). The angle between ru and I’ is Q,. The position vector sweeps out a circle in the primed coordinate frame, where angle a is defined relative to the x’-axis. Calculation of the scattering pattern rquires a triple-integration over a,tj, and Q. To simplify the illustration, the x’and y’axes are shown intersecting at z’ = ru co8 Q, and not at z’ 0.

-

TABLE 1: (Ref 36).

Structural Parameters for Trifluoromethyl Iodide

no. r,lA hJlA Qijldeg 0.0510 110.6 3 1.3314 0.0465 0.0 1 2.1409 90.0 3 2.1535 0.0547 F--I 3 2.8916 0.08 12 154.5 a Internuclear separation, ru, mean amplitude of vibration, 11,; angle between rij and C-I axis, Q,. atom pairs C-F c-I F-F

The CF3 radicals will take on a perpendicular/excited distribution at each rotational recurrence; since the moment of inertia ofCF3 is not equal to the moment of inertia of CF31, the rotational recurrence times of CF3 will be different (and resolvable) from the parent species.

IV. Vibrational Coherence In the previous section it was shown that additional structural and spectroscopic information can be obtained by observing the diffraction pattern of a laser-excited molecular sample during rotationalrecurrences. The experimental factors that enable this are the ultrafast laser pulse, which creates a coherent superposition of rotational states, and the ultrafast electron pulse, which has sufficient time resolution (a few picoseconds) to record the diffraction pattern of the molecular sample during the rotational recurrence window. Here we investigate the electron scattering patterns from a molecular beam sample with vibrational coherence. A femtosecond laser pulse creates the coherence, a superposition of vibrational eigenstates. Traditionally, the time evolution of this superposition, or wave packet, is monitored by a second femtosecond laser pulse with FTS to follow the molecular dynamics of unbound, bound, and quasi-bound systems.17 We wish to show that UED provides similar information about the dynamics and potential energy surfaces of molecular systems; the simulations presented are important to test two regimes for- structural changes: coherent motion and the kinetic regime. UED may be conducted even when little spectroscopic information is known about the system-the probe involves only the state of interest, and not a transition involving two states. Furthermore, the dynamics along the reaction coordinate can be extracted for complex systems because UED records all structural changes. In

1116 The Journal of Physical Chemistry, Vol. 98, No. I I , 1994

Williamson and Zewail

(a)

4.0

4.0

0

1

2

3

4

Internuclear Separation (A) Figure 14. Radial distribution curve cross sections of isotropic CFjI and of the perpendicularfunexcited CFjI subset. The curves have k e n baseline-corrected to emphasize thechanges in each pcak (wmpare with the uncorrected surface in Figure 13b). As discussed in the text, the C-I and F-I pcaks arc most intcnse along the y-axis and least intense along the x-axis. The C-F and F-F peaks are most intense along the x-axis.

Internuclear Separation

(A)

Figure 13. Two-dimensional radial distribution curves of CF& (a) Isotropic sample. The inner ring wrresponds to the C-F separation, the middle ring is a sum of the C-I and F-F separations, and the outer ring wrresponds to the F-I separation. Only one quadrant is shown. (b) Perpendicularfunexcited subset of the molecular sample after laser excitation along the C-I band axis. Only one quadrant is shown. The inner ring is most intense along thc x-axis because ~ G F is almost perpendicular to rc.1. The outer ring is most intense along the y-axis

becauserF-Iisalmost parallel torc.1. The twocomponentsofthemiddle ring have been separated along the x- and y-axes because rF.+ IrGI. Scattering from TCI is found along the y-axis. This peak is less intense than therp-Fpeakalong thex-axis becausescatteringfromrc-lisdamped by thews(qc-s) term. Distortions in the baseline (seesection II1,part D) actually exaggerate this effect. but it can still be ObseNed in the corrected profiles (Figure 14) this section we will comparethediffraction patterns from electron pulse/velocity mismatch temporal resolutions (re)of both 50 and 500 fs. The experimental limit of time resolution is discussed in the accompanying paper.21 The diffraction patterns are determined as a function of time by calculating the electron scattering from classical trajectories ofmolecular iodineexcited totheBstate. Asaresult,thissection begins with a discussion of the scattering equations involved in trajectory calculations, and the trajectory calculations themselves are then described in part B. We discuss the diffraction patterns when B-state iodine dissociates (part C) and when it is trapped within the potential well (part D), and conclude this section by examining the experimental applications of UED to molecular dynamics studies on several different time.scales. A. Theoretical Framework. To emphasize the role of vibrational coherence in the diffraction patterns, we will ignore the effects of anisotropic spatial orientation distributions in the

scattering sample and assume that the excited molecules have an isotropic distribution The diffraction patterns will therefore be cylindrically symmetric and consist of sums of the zero-order spherical Bessel function, jo. This will a m u n t for the rotational averaging shown in eq 5 , but vibrational averaging must be considered more carefully. Our classical trajectory calculation consists of a molecular ensemble in which the initial conditions of each trajectory are randomly determined with a Monte Carlo technique according to some probability distribution, The total number of trajectories, NT. must he sufficiently large such that the macroscopic behavior emerges by averaging the behavior of the individual trajectories. Consequently, if the molecular scattering curve is calculated from a large numberoftrajectories, than the total scattering should automatically a m u n t for vibrational averaging. Themolecular scattering intensity fromaset ofNTtrajectories for molecular iodine simplifies to

where r.(f) is the evolution of the internuclear separation of the nth trajectory in time. Two-dimensional diffraction plots with the modified molecular scattering along the abscissa (in A-I) and time along the ordinate (in fs) will be created. Generating radial distribution curves from these plots is straightforward. B. Molecular Dynamics Calculations. With molecular dynamics simulations of solutions, Bergsma et al. have shown the calculated Vibrational dynamics of iodine in different solvents following X-ray diffraction.'6 Here, the trajectory calculation was designed to mirror the UED experimental excitation process, where the pump laser excites isolated molecular iodine, creating a wave packet whichevolvesin timeon theB state. Thisevolution was recorded for each trajectory inaone-dimensional array,r.(f), and the diffraction pattern at a specific time 1, was obtained by summing the scattering contributions from all rn(f,) (eq 32). We wish to compare the diffraction patterns from several different excitation energies, 2000,3000, and 4000 cm-I, which are below the dissociation energy, and a t 4750 cm-I, which is above the dissociation energy. These energies are relative to the bottom of the B-state potential One thousand trajectories were used for the bound systems, and tenthousandtrajectorieswereusedforthedissociativesystem. It wasassumed that theinitialground-state molecular iodinewas

Ultrafast Electron Diffraction I

(a)

~

, I

i

.

The Journal of Physical Chemistry, Vul. 98. Nu. 11,1994 2111

..~~ \-._ ~

-

10

Internuclear

10

s

(A')

/

Separation (A) Figure 15. Molecular dynamics of B-state iodine excited to 4750 cm-I above the bottom of the potential well. Iodine dissociates at this energy. Electron pulse temporal resolution: r. = 50 fs. (a) Modified molecular scattering curve as a function of time. (b) Radial distribution curve as a function of time. The hump at 400 fs appears because the wave packet slows dawn as it climbs the outer potential wall before dissociation. vibrationally cold, with the initial energy and probability distributions corresponding to Y = 0. The excitation laser pulse had a Gaussian temporal profile (fwhm = 50 fs), and the energy bandwidth was determined by assuming that the pulse was transform limited. The goal of the trajectory calculation was to generate a twodimensional array containing internuclear separation probabilities as a function of time. The size of each position element was 0.01 A and the size of each temporal element was 5 fs. The initial conditions of each trajectory were determined using the Monte Carlo technique. The initial energy was equal to the excitation energy (e.&, 3000 cm-I) plus an offset determined by the laser bandwidth, and the initial position was determined from the probability distribution ofground-stateidine. Several iterations were sometimes necessary to establish a valid combination of initial energy and position, particularly a t lower energies where the Franck-Condon overlap between the X state and the B state is poor. The initialvelocitywas calculated from the initial energy and the B-state potential surface, and a randomly determined direction for the velocity (Le., bond contracting or expanding) was assigned. After defining the initial conditions, the evolution of each trajectory on the B-state potential surface was calculated. The program employed a Hulburt-Hirschfelder potential energy surface" with Dunham coefficients provided by Gerstenkorn and

0

2

4

6

8

1

0

Internuclear Separation (A) Figure 16. Molecular dynamics of B-state iodine excited to 4750 Em-' above the bottom of the potential well. Iodine dissociates at this energy. Electron pulsetemporal resolution: 7. = 500 fs. (a) Modified molecular scatteringcurveas a functionof time. Theascillationsdampoutquickly because of the low temporal resolution. (b) Comparison of the radial distribution CUNS after I ps for = 50 fs and 7, = 500 fs. The radial distribution C U N ~at time zero ( T ~= 50 fs) is shown as a refercnce. L ~ c . 3New ~ positions and velocities were calculated every 5 fs using an embedded Runge-Kutta integration scheme designed to conserve the total energy of each traject0ry.3~The calculation began a t 4 . 5 ps (with the static ground-state distribution) and continued out to +2.5 ps. The zero of time, f = 0, corresponded to excitation of the molecular sample with the pump laser. Theduration of the pumplaser pulse was included by offsetting time zero for each trajectory by a random amount determined witha MonteCarlotechniqueon thepumplasertemporal profile. Prior to to.., the two-dimensional array was filled with the probability distribution ofground-state iodine normalized to unit area. Evolution of the trajectory after was modeled by adding one to the array element corresponding to the current time and calculated position. The temporal profile of the electron pulse (including velocity mismatch effects) was assumed to be Gaussian. The duration of the electron pulse was included in the calculation by convoluting the temporal behavior of a specific internuclear distance (one row in the array) with the temporal profile of the electron pulse. The final result was an array of internuclear distances ranging in time from 0 to 2 ps. The modified molecular scattering curve was calculated from 0 to 15 A-' (in 0.02-A--1 steps) at each time interval using eq 31, and the radial distribution curve was calculated from sM(s) with kd = 0.01 A2.

2118

Williamson and Zewail

The Journal of Physical Chemistry, Vol. 98, No. 11. 1994 ,

(a)

1

._ -.__ ,

..

'-,

' . .

Internuclear Separation (A)

Internuclear Separation (A)

(c)

-. . 1 -

//

,/

(d)

L.\\..,. -. 1..

\~

...~.

Internuclear Separation (A)

Internuclear Separation (A) Figure 17. Molecular dynamics of B-state iodine excited below the dissociation energy (4381.8 cm-I): (a) 2000 cm-'. = 50 fs; (b) 3000 cm-', 1. = 50 fs; (c) 4000 cm-1, *I = 50 fs; (d) 4000 cm-1, . T = ~ 500 fs. In all graphs, the peak intensity at the inner and outer turning pints decreases as a function of time, which shows that the classical wave packet is spreading and approaching a Static distribution. curve for re = 500 fs in Figure 16a. The interference fringes C. Above theDissociationEnergy. The first systemof interest are almost completely washed out after the first 400 fs and the was iodine excited to 4750 cm-1 above the bottom of the B-state notentialwell. ThedissociationthresholdoftheB-stateislocated molecularscatterineintensitvisvervsmall. Acomoarison ofthe at 4381.8 cm-l, and it is expected that the diffraction patterns radial distribution curves a t 1 ps for re = 50 fs and re = 500 fs will reflect a continuous increase in the internuclear separation (Figure 16b) shows that the peak flattens significantly as the temporal resolution changes by an order of magnitude. of the molecule with time. The modified molecular scattering D. Below the Dissociation Energy. We also performed curve and radial distribution curves with re = 50 fs are presented moleculardynamicscalculationsfor idineatthreeenergies (2000, in Figure 15. The radial distribution curve shows that iodine 3000,and4000c11-~)below thedissociationenergyoftheB-state. molecules traverse the potential well within the first 500 fs and The radial distribution curves are plotted as a function of time subsequently dissociate. The position of the radial distribution in Figure 17. When the temporal resolution is high (7. = 50 fs), peak increases in distance with time, and the peak also flattens thevibrational motion is clearly present, even for fast oscillations out as the wave packet gradually disperses across the potential at 2OOOcm-1. Closeexamination shows thatvibrationaldephasing surface. The modified molecular scattering curve presents the occurs, and one can expect that vibrational quantum coherence same information, but in the inverse-distance domain. The would be apparent if a quantum mechanical wave packet frequency of oscillation increases with time as 11.1 increases. The calculation was conducted. As with FTS,a Fourier analysis of amplitude of the oscillations diminish rapidly as the wave packet this oscillatory motion would yield the spacing between the spreads; scattering contributions from a wide range of internuclear Vibrational energy levels.41 separations lead to a low scattering intensity. The oscillation period a t 4000 cm-I is long enough that the The calculated behavior of iodine is similar to observations of vibrational motion may still be resolved when re = 500 fs (Figure dissociating I2 and ICN made with FTWMand to the corres17d). although there appears to be rapid dephasing. The ponding quantum mechanical wave packet calculations of the vibrational motion a t lower energies is not resolved, and the FTS ex~eriment.".~2 UED Drovides the actual dvnamical diffraction pattern is static after t h e first 500 fs. This static trajectory of the fragments. Aiso, by taking diffraction patterns pattern represents a weighted sum of the position probability at various times after laser excitation and calculating the change amplitudes of the vibrational states excited within the laser in the internuclear separation, it would he possible to determine bandwidth, and therefore these diffraction patterns are useful the kinetic energy distribution of the fragments and deduce the for determining the shapeofthe potentialenergysurface. Figure dissociation energy. Such measurements require femtosecond 18 shows the 2-ps radial distribution functions (with rC= 500 fs) time resolution, however, as is shown by the molecular scattering $I

-

Ultrafast Electron Diffraction 5000

The Journal of Physical Chemistry, Vol. 98, No. 1I, 1994 2779

I

4000

c E 3000

2

0I

*

r

C

M 4 0 6 0 8 0 1 0 0

I

Time (DS)

2000

w

1000

+--c--cI

1

2

3

4

5

6

Internuclear Separation (A) Figure 18. Radial distribution curves at 2 ps from molecular dynamics of B-state iodine excited below the dissociation energy. The RD curves are shown superimposed over the B-state potential energy surface. Electron pulse temporal resolution: T~ = 500 fs. At this temporal resolution, the trajectories for 2000 and 3000 cm-l are static after the first 500 fs, although vibrational oscillations of B-state iodine molecular dynamics at 4000 cm-I may still be resolved (see Figure 17d).

for all three energies overlaid on the B-state potential energy surface. Note that the curves for 2000 and 3000 cm-1 are static and represent the average vibrational motion, but the 4000 cm-1 curve is still evolving in time. This type of experiment has no need of ultrafast time resolution and is actually better suited to longer time scales in which the bandwidth of the excitation laser is narrow enough to pick out specific vibrational states-state selective diffraction. E. Experimental Applications to More Complicated Systems. Here, we discuss ways UED might be applied to dynamical experiments where reaction intermediates are formed on longer time scales. The most important characteristic of gas-phase electron diffraction is that the scattering patterns are extremely sensitive to small changes in internuclear separation. Conventional GED experiments typically report internuclear separations to within 0.005 or less;4 so far, a limited range of s is detected

inUED(lSA--',asopposedto -4OA-linGED),andtheprecision is more on the order of 0.05 A. This sensitivity is still sufficient to distinguish between different vibrational states (Figure 18) as well as between different electronic states (Figure 8). UED therefore provides another detection scheme for monitoring ultrafast decay processes, either when the products dissociate or when they merely change to a different vibrational or electronic state, Such an avenue of detection is, of course, particularly useful when spectroscopic data about higher excited states is not available or is too complicated. As an example of how UED might be applied to dynamical studies, we can consider the three-body photofragmentation process of CzF412 studied in real time.43 The photofragmentation is a two-step process in which both iodine atoms dissociate, leaving tetrafluoroethylene

'I

+

C2F412* C2F41* I* '2

C2F41*+ F2C=CF2

+I

(33) (34)

The primary C-I bond breakage is very fast ( r l 5 0.5 ps), while the secondary bond breakage results from internal energy redistribution and is significantly slower (72 = 32 ps at a given energy). The dynamics of the two steps depended on the total energy and were probed by measuring the rises of excited-state atomic iodine (I*) and ground-state atomic iodine (a biexpo-

Internuclear Separation (A)

5

6

Figure 19. Reaction dynamicsof CzF412. Thefirst iodine atom dissociates directly with 71 I0.5 ps. The second iodine atom dissociates with 7 2 = 32 ps. (a) Ground-state iodine atom signal as a function of time (ref 43). (b) Radial distribution curve as a function of time. At time zero the molecular sample is pure C2F412. After time zero the molecular sample is a mixture of C2F.J radical and F2C=CF2. The composition was obtained from the experimental results for 71 and 72.

nential; see Figure 19a). The corresponding time evolution of

theradialdistributioncurveisshowninFigure 19b. Themolecular scattering curve is simply a weighted sum of the three components (C2F412, C2F41,and F&=CF2); the composition of the molecular sample was calculated as a function of time using measured values of T~ and 72. Structural parameters for C2F4I2 and F2C==CF2 were obtained from the l i t e r a t ~ r e ~and ~ pthe ~ ~structure of the radical component was assumed to be identical to C2F4I2 with one iodine atom missing. The radial distribution curve clearly reveals the dynamical changes of the fragmentation process. The primary bond breakage is shown by the rapid disappearance of the peak at 5.0 A, which corresponds to the I-.I internuclear separation. The subsequent transformation of the radial distribution curve over the next 100 ps characterizes the secondary bond breakage; note that the intensity of the first peak at 1.3 A (C-C, C-F) does nor change because all carbon and fluorine atoms remain in the molecule, although the position of the peak shifts inward slightly as the C-C bond changes to a double bond. The ability of UED to monitor this type of dynamical change has already been demonstrated19 and is detailed in the accompanying paper.2' The primary and secondary bond breakages in C2F4I2 are very different processes and help illuminate the distinctions between monitoring vibrational coherence versus following dynamical decays. Loss of the first iodine atom is a direct process and very fast; if the molecule is excited with a femtosecond laser pulse, a wave packet is formed on the dissociative potential energy surface. With a 50-fs electron pulse, it would be possible to monitor this dissociation process in time, and the diffraction patterns would be similar to those shown in Figure 15. The 1-1 separation would rapidly move outward from 5.0 A, and one set of C-I, C-I, and Fa-I peaks would shift outward as well. Loss of the second iodine atom, however, is not a coherent process. The individual C2F41 radicals dissociate independently of each other, with the behavior of the entire population characterized by the decay constant 7 2 . Trajectory calculations are unnecessary to model this type of dynamical process because the loss of the second iodine is very fast compared to the uncertainty in the onset of the dissociation.

Williamson and Zewail

2780 The Journal of Physical Chemistry, Vol. 98, No. 11, 1994

TABLE 2: Summary of Spatial Scattering Functions F, for Several Spatial Orientation Probability Distributions fin,$) Where

n,=

0.

spatial orientation distribution isotropic

fraction of molecules 1

P(Q,+)

Fij(5)

Jo[sr,jl

1

parallel excited

'13

cos2 a

parallel unexcited

2/ 3

sin2 Q

perpendicular excited

'I3

sin2 f? cos2

perpendicular unexcited

2 13

1- sin2Q cos2

+

1 - sin2Q sin2IL

unpolarized excited

unpolarized unexcited

+

Ii3

sin2 f? sin2

+

perpendicular excited

S'ij

(

sin2 Qij

+ (2-3 sin2

Q,j)

The bottom row shows the functional form of the perpendicu1arJexcited Fij for arbitrary Thus the diffraction pattern may be calculated as a sum of scattering from each of the three components weighted by their relative populations.

acknowledges the National Science Foundation for providing a Graduate Student Fellowship.

Appendix V. Conclusion Debye's work on X-ray diffraction showed that the scattering pattern from a gas is rich with structural information, despite the randomness in the position and orientation of the molecular sample. One focus of this paper was to show that ultrafast preparation and ultrafast electron diffraction can be used to introduce and observe order in gas phase samples, thus yielding a new dimension in molecular structure analysis. The formation of a coherent superposition of rotational and vibrational eigenstates, which can be controlled by the experimental time delay, are transient impositions of this order on an otherwise isotropic sample. Rotational recurrence UED provides a detailed structural picture of the ground-state species: internuclear distances, moment of inertia (from the rotational constant), and bond angles. Vibrational coherence studies reveal transition-state dynamics and provide a map of the potential energy surface; this information is very similar to results from FTS. Applications were discussed for small systems (two atoms), more complex systems (five or eight atoms), and reactions involving direct dissociation or reactive intermediates. The following paper details our apparatus, its temporal and spatial resolutions, and experimental applications.

Acknowledgment. This work was supported by a grant from the U S . Air Force Office of Scientific Research and the National Science Foundation. The authors thank Dr. Scott Kim and Dr. Marcos Dantus for their stimulating discussions on ultrafast electron diffraction, and Hakno Lee and Dr. Juen-Kai Wang for their help. We thank Dr. Ralf Friichtenicht for assisting in the translation of some of the referenced (German) articles. J.C.W.

Many of the definite integrals generated from eq 17 are not readily available in mathematical references. This appendix provides a general description of the integration process needed to obtain the closed form solutions summarized in Table 2. Integration of eq 17 over J, is accomplished with one of the following identities:

or

(A-2) where y is defined by y = (A2

+ B2)'/*

('4-3)

Equation A-1 may be verified by expanding the exponential as a series and employing definite integrals listed in Gradshteyn and Ryzhik.468 Equation A-2 may be verified by expanding the exponential, expanding each term in the exponential sum, and integrating term by term.6b As anexampleof how theseidentities are applied, eq A-1 simplifies F',,for the isotropic distribution to

F,,(s) = iJ:sin

R Jo[ sr cos sin R] eisrsin(e/2)oos* dR (A-4) 2

The final closed form is obtained by integrating over R using one of the following identities:

The Journal of Physical Chemistry, Vol. 98, No. 11, 1994 2781

Ultrafast Electron Diffraction Jor(sin (p)"+'J,,[Bsin v]eiAcoscP dv = 2 --j,[y] B" Y

(A-5)

or Jorcos2 cp(sin cp)"+'J,[B sin cp]eAmcPdv =

These integrals may be verified by expanding the Bessel function as a series and making a change of variables to t = cos cp. Integration over t generates another Bessel function,* which is also expanded. The double expansion that results for eq A-5 is

and the double sum for eq A-6 is similar. The final step is to change the summation limits by defining i = m k and j = k

+

(13) Mourou, G. A.; Williamson, S . Appl. Phys. Lett. 1982, 41, 44. Williamson, S.;Mourou, G.; Li, J. C. M. Phys. Rev. Lett. 1984, 52, 2364. (14) Elsayed-Ali, H. E.; Herman, J. W. Reo.Sci. Instrum. 1990,61,1636. Herman, J. W.; Elsayed-Ali, H. E.; Murphy, E. A. Phys. Rev. Lett. 1993,71, 400. (15) Kitriotis, D.; Aoyagi, Y. Jpn. J. Appl. Phys. 1993, 32, L441. (16) Bergsma, J. P.; Coladonato, M. H.; Edelsten, P. M.; Kahn, J. D.; Wilson, K. R.; Fredkin, D. R. J. Chem. Phys. 1986,84,6151. (17) Zewail, A. H. Science 1988,242, 1645. Khundkar, L. R.; Zewail, A. H. Annu. Rev. Phys. Chem. 1990,41,15. Zewail, A. H. Faraday Discuss. Chem. SOC.1991,91,207. Zewail, A. H. J. Phys. Chem. 1993,97, 12427. (18) Williamson, J. C.; Zewail, A. H. Proc. Natl. Acad.Sci. U.S.A.1991, 88, 5021. (19) Williamson, J. C.; Dantus, M.; Kim, S. B.; Zewail, A. H. Chem. Phys. Lett. 1992, 196, 529. (20) Williamson. J. C.: Zewail. A. H. Chem. Phvs. Lett. 1993.209. 10. (21) Dantus, M.; Kim, S. B.; Williamson, J. C.;-Zewail, A. Hi J . Phys. Chem., following paper in this issue. (22) Karle, J. In Determination of OrganicStructures by Physical Meihods; Nachod, F. C., Zuckerman, J. J., Eds.;Academic Press: New York, 1973; P 1. (23) Schiifer, L. Appl. Spectrosc. 1976, 30, 123. (24) Yates, A. C. Phys. Rev. 1968,176, 173. (25) Ibers, J. A,, Hamilton, W. C., Eds. International Tables ForX-Ray Crystallography; Kynoch Press: Birmingham, 1974; Vol. 4, p 176. Sellers, H. L.; Schiifer, L.; Bonham, R. A. J. Mol. Struct. 1978,49, 125. (26) Felker, P. M.; Zewail, A. H. J. Chem. Phys. 1987,86, 2460. (27) Baskin, J. S.;Felker, P. M.; Zewail, A. H. J . Chem. Phys. 1987,86, 2483. Baskin, J. S.;Zewail, A. H. J. Phys. Chem. 1989, 93, 5701. Felker, P. M.;Zewail, A. H. In Femtosecond Chemistry; Manz, J., Wbste, L., Eds.; VCH: New York, 1994. Baskin, J. S.;Zewail, A. H. J . Phys. Chem. 1994, in..aress. .

Summation over j leads to a y2i term, and the sum over i corresponds to the final Bessel function.

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(28) Kohl, D. A.; Shipsey, E. J. Z . Phys. D 1992, 24, 3 3 . Kohl, D. A.; Shipsey, E. J. Z . Phys. D 1992, 24, 39. (29) Fink, M.: Ross, A. W.; Fink. R. J. Z . Phvs. D 1989,ll. 231. Mihill. A.;'Fink, M. Z . Phys. D 1989, 14, 77. (30) Volkmer, M.; Meier, Ch.; Mihill, A.; Fink, M.; BBwering, N. Phys. Rev. Lett. 1992, 68, 2289. (31) Note that Fink et al initially define +relative to the x-axis in Figure 1 of their first paper (ref 29). It appears, however, that is defined relative to the y-axis in their text and in subsequent articles. (32) Spiridonov, V. P.; Gershikov, A. G.; Butayev, B. S.J. Mol. Struct. 1979, 51, 137. (33) Halliday, D.; Resnick, R. Physics, 3rd ed.; J. Wiley & Sons: New York, 1978; p 994. (34) Luc, P. J. Mol. Spectrosc. 1980, 80, 41. (35) Bowman, R. M.; Dantus, M.; Zewail, A. H. Chem. Phys. Lett. 1989, 161; 297. (36) Typke, V.; Dakkouri, M.; Oberhammer, H. J . Mol. Struct. 1978.49, 85. (37) Hulburt, H. M.; Hirschfelder, J. 0. J. Chem. Phys. 1941, 9, 61. Hulburt, H. M.; Hirschfelder, J. 0. J . Chem. Phys. 1961, 35, 1901. (38) Gerstenkorn, S.;Luc, P. J. Phys. (Paris) 1985, 46, 867. (39) Press, N. H.; Teukolsky, S.A.; Vetterling, W. T.; Flannery, B. P. Numerical Recipes In FORTRAN, 2nd ed.; Cambridge University Press: Cambridge, 1992; p 701. (40) Dantus, M.; Rosker, M. J.; Zewail, A. H. J. Chem. Phys. 1988,89, 6128. (41) Gruebele, M.; Zewail, A. H. J. Chem. Phys. 1993, 98,883. (42) Williams, S.0.;Imre, D. G. J . Phys. Chem. 1988, 92,6648. (43) Knee, J. L.; Khundkar, L. R.; Zewail, A. H. J. Chem. Phys. 1985, 83, 1996. Khundkar, L. R.; Zewail, A. H. J. Chem. Phys. 1990, 92, 231. (44) Thomassen, H.; Samdal, S.;Hedberg, K. J. Am. Chem. SOC.1992, 114,2810. (45) Carlos, J. L.; Karl, R. R.; Bauer, S. H. J. Chem. Soc., Faraday Trans. 2 1974, 70, 177. (46) Gradshteyn, I. S.;Ryzhik, I. M. Tables Of Integrals, Series, and Products; Academic Press: New York, 1980 (a) p 382, eq 3.661-1.2; (b) p 131, 2.512; (c) p 321, 3.387-2.

+